- #1
jetplan
- 15
- 0
Hi All,
So all closed interval [a,b] is compact
(see Theorem 2.2.1 in Real Analysis and Probability by RM Dudley)
Now, Let's say I have [0,10] as my closed interval.
Let My Open Cover be
(0, 5)
(5, 7.5)
(7.5, 8.75)
(8.75, 9.375)
...
Essentially, The length of each open interval is cut by half, i.e.
length (0,5) = 5
length (5,7.5) = 2.5 = 5/2
length (7.5,8.75) = 1.25 = 5/(2*2)
length (8.75, 9.375) = 0.625 = 5/(2*2*2)
...
the Union of all these interval gives us [0.10] \ ({0} U {5} U {7.5} U {8.75} U {9.375} ... )
Therefore, we add
(0-[tex]\epsilon[/tex], 0+[tex]\epsilon[/tex])
(5-[tex]\epsilon[/tex], 5+[tex]\epsilon[/tex])
(7.5-[tex]\epsilon[/tex], 7.5+[tex]\epsilon[/tex])
(8.75-[tex]\epsilon[/tex], 8.75+[tex]\epsilon[/tex])
... to cover all these missing single points.
The choice of [tex]\epsilon[/tex] is arbitrary, as long as it doesn't touch the middle-point of the next interval.
for example,
to cover {5}, we can add a (5-0.1, 5+0.1)
to cover {7.5} we can add a (7.5-0.01, 7.5+0.01)
etc etc
So we have create ourselves an Open cover for [0, 10]
but i can't see any finite members of such cover become yet another cover for [0, 10]
I know [0, 10] is compact, I simply can't find a finite subcover.
Where is the loophole ?
Thanks
J
So all closed interval [a,b] is compact
(see Theorem 2.2.1 in Real Analysis and Probability by RM Dudley)
Now, Let's say I have [0,10] as my closed interval.
Let My Open Cover be
(0, 5)
(5, 7.5)
(7.5, 8.75)
(8.75, 9.375)
...
Essentially, The length of each open interval is cut by half, i.e.
length (0,5) = 5
length (5,7.5) = 2.5 = 5/2
length (7.5,8.75) = 1.25 = 5/(2*2)
length (8.75, 9.375) = 0.625 = 5/(2*2*2)
...
the Union of all these interval gives us [0.10] \ ({0} U {5} U {7.5} U {8.75} U {9.375} ... )
Therefore, we add
(0-[tex]\epsilon[/tex], 0+[tex]\epsilon[/tex])
(5-[tex]\epsilon[/tex], 5+[tex]\epsilon[/tex])
(7.5-[tex]\epsilon[/tex], 7.5+[tex]\epsilon[/tex])
(8.75-[tex]\epsilon[/tex], 8.75+[tex]\epsilon[/tex])
... to cover all these missing single points.
The choice of [tex]\epsilon[/tex] is arbitrary, as long as it doesn't touch the middle-point of the next interval.
for example,
to cover {5}, we can add a (5-0.1, 5+0.1)
to cover {7.5} we can add a (7.5-0.01, 7.5+0.01)
etc etc
So we have create ourselves an Open cover for [0, 10]
but i can't see any finite members of such cover become yet another cover for [0, 10]
I know [0, 10] is compact, I simply can't find a finite subcover.
Where is the loophole ?
Thanks
J